Second, our family of groups contains the groups \(\mathcal {B}_g\)  of Aramayona–Funar–Kapoudjian [1]}, [2]}, [3]}. More concretely, the group
\(\mathcal {B}_0\)  of [1]} is \(\mathcal {B}_{2,1}(\mathbb {S}^2, \mathbb {S}^2)\) ; the group \(\mathcal {B}_g\)  of [3]} is \(\mathcal {B}_{2,1}(S_g, \mathbb {S}^2)\) , where \(S_g\)  denotes the closed surface of genus \(g\) ; and the group \(\mathcal {B}_\infty \)  of [2]} is \(\mathcal {B}_{2,1}(\mathbb {S}^2, \mathbb {S}^1 \times \mathbb {S}^1)\) ; see Proposition REF . Thus we have the following affirmative answer to [7]} and [8]}:
